If that's the only one you want to train at then don't bother until you get your license. You'd be asking your parents to make an hour trip to drive you there and then drive themselves home, and then another hour trip to pick you up and come home. I had a similar fantasy in regards to joining a Muay Thai gym which was a similar distance from home. Just forget about it and do Judo in the meantime. You won't regret it.
It occurs to me that perhaps explaining how real boxing is might be the wrong way to go.
Instead, the OP may wish to point out that in karate they can punch AND kick, while in boxing they only punch. Further, there is the protection of gloves and headgear and groin cups and mouthguards, while in karate nothing stands between someone and DEATH but some foreign pajamas. Sure, it's 8/9ths kata or whatever, but what about that other ninth, mom?! What about that other ninth?!?!?
Tell your mom that karate just isn't doing it for you any more. You used to be at full mast after a quick gedan barai, but now you can't even feel anything during a basadai. You've just changed from when you first met karate, and you're in a different place in your life. It isn't karate, its you. And it isn't fair to karate to keep staying with it just for the stake of staying with it. Karate and you can still be friends; you know, go out and point spar every now and again. But what you used to have is gone. It's time to move on from karate.
Lol. I'm gonna take Rask's advice for now, until I move out.
In the meantime, I just found out that my friend's dad has experience with boxing (and corrected my jab/oi zuki as of about 20 minutes ago), as does my school PE teacher. So, I can get the basics from them, and go and apply what I've learned full contact with my friends until I can actually start fighting with real fighters.
I jumped from page 1 to 7, so if this has already been suggested - oh well.
How are your grades?
If so-so, tell mom you'll improve to show her how serious you are.
Offer to help earn money for your needs.
I got me a mini bike after much effort. Then came more stitches.
You could always tell her that "what doesn't kill you makes you stronger". That one didn't work, but maybe your mom is different.
Best yet - go with the pack and just start judo. You'll never regret it.